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Handout Monday 11/10
We have claimed that the key property that a bijection between two groups must have for it to be an isomorphism should be that \(\varphi(ab) = \varphi(a)\varphi(b)\text{.}\) For rings, we must “respect” both operations, so \(\varphi(a+b) = \varphi(a)+\varphi(b)\) and \(\varphi(ab) = \varphi(a)\varphi(b)\text{.}\)
But does this really do the job of ensuring that the two groups are “basically the same”?
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Prove that under any isomorphism, the identity must be sent to the identity.
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Prove that under any isomorphism, the inverse of an element must be sent to the inverse of its image.
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Prove that if \(\varphi:G_1 \to G_2\) is an isomorphism, then \(G_1\) is abelian if and only if \(G_2\) is abelian.
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What if \(\varphi:R_1 \to R_2\) is a ring isomorphism. Prove that \(R_1\) is an integral domain if and only if \(R_2\) is an integral domain.
